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(Pseudo) Random Quantum States with Binary Phase
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| Abstract: | We prove a quantum information-theoretic conjecture due to Ji, Liu and Song (CRYPTO 2018) which suggested that a uniform superposition with random binary phase is statistically indistinguishable from a Haar random state. That is, any polynomial number of copies of the aforementioned state is within exponentially small trace distance from the same number of copies of a Haar random state.As a consequence, we get a provable elementary construction of pseudorandom quantum states from post-quantum pseudorandom functions. Generating pseudorandom quantum states is desirable for physical applications as well as for computational tasks such as quantum money. We observe that replacing the pseudorandom function with a (2t)-wise independent function (either in our construction or in previous work), results in an explicit construction for quantum state t-designs for all t. In fact, we show that the circuit complexity (in terms of both circuit size and depth) of constructing t-designs is bounded by that of (2t)-wise independent functions. Explicitly, while in prior literature t-designs required linear depth (for $$t > 2$$), this observation shows that polylogarithmic depth suffices for all t.We note that our constructions yield pseudorandom states and state designs with only real-valued amplitudes, which was not previously known. Furthermore, generating these states require quantum circuit of restricted form: applying one layer of Hadamard gates, followed by a sequence of Toffoli gates. This structure may be useful for efficiency and simplicity of implementation. |
BibTeX
@article{tcc-2019-29974,
title={(Pseudo) Random Quantum States with Binary Phase},
booktitle={Theory of Cryptography},
series={Lecture Notes in Computer Science},
publisher={Springer},
volume={11891},
pages={229-250},
doi={10.1007/978-3-030-36030-6_10},
author={Zvika Brakerski and Omri Shmueli},
year=2019
}